Mechanical systems with nonholonomic constraints

Abstract

A geometric setting for the theory of first-order mechanical systems subject to general nonholonomic constraints is presented. Mechanical systems under consideration are not supposed to be Lagrangian systems, and the constraints are not supposed to be of a special form in the velocities (as, e.g., affine or linear). A mechanical system is characterized by a certain equivalence class of 2-forms on the first jet prolongation of a fibered manifold. The nonholonomic constraints are defined to be a submanifold of the first jet prolongation. It is shown that this submanifold is canonically endowed with a distribution—this distribution (resp., its vertical subdistribution) has the meaning of generalized possible (resp., virtual) displacements. The concept of a constraint force is defined, and a geometric version of the principle of virtual work is proposed. From the principle of virtual work a formula for a workless constraint force is obtained. A mechanical system subject nonholonomic constraints is modeled as a deformation of the original (unconstrained) system. A direct characterization of a constrained system by means of a class of 2-forms along the canonical distribution is given, and “constrained equations of motion” in an intrinsic form are found. A geometric definition of regularity for systems under nonholonomic constraints is provided. In particular, the case of Lagrangian systems is discussed. Also systems subject to holonomic constraints and nonholonomic constraints affine in the velocities are investigated within the range of the general scheme.